let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being Program of SCM+FSA st I is_halting_on Initialized s,P holds
( ( for a being read-write Int-Location holds (IExec (I,P,s)) . a = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . a ) & ( for f being FinSeq-Location holds (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f ) )
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_halting_on Initialized s,P holds
( ( for a being read-write Int-Location holds (IExec (I,P,s)) . a = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . a ) & ( for f being FinSeq-Location holds (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f ) )
set A = NAT ;
let I be Program of SCM+FSA; :: thesis: ( I is_halting_on Initialized s,P implies ( ( for a being read-write Int-Location holds (IExec (I,P,s)) . a = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . a ) & ( for f being FinSeq-Location holds (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f ) ) )
set s0 = Initialized s;
set s1 = Initialize (Initialized s);
set P1 = P +* I;
assume I is_halting_on Initialized s,P ; :: thesis: ( ( for a being read-write Int-Location holds (IExec (I,P,s)) . a = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . a ) & ( for f being FinSeq-Location holds (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f ) )
then A2: P +* I halts_on Initialize (Initialized s) by SCMFSA7B:def 8;

hereby :: thesis: for f being FinSeq-Location holds (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f

let a be read-write Int-Location ; :: thesis: (IExec (I,P,s)) . a = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . a
not a in NAT by SCMFSA_2:84;
then not a in (dom s) /\ NAT by XBOOLE_0:def 4;
then A3: not a in dom (s | NAT) by RELAT_1:90;
thus (IExec (I,P,s)) . a = ((Result ((P +* I),(Initialize (Initialized s)))) +* (s | NAT)) . a by SCMFSA8A:13
.= (Result ((P +* I),(Initialize (Initialized s)))) . a by A3, FUNCT_4:12
.= (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . a by A2, EXTPRO_1:23 ; :: thesis: verum

end;

let f be FinSeq-Location ; :: thesis: (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f
not f in NAT by SCMFSA_2:85;
then not f in (dom s) /\ NAT by XBOOLE_0:def 4;
then A4: not f in dom (s | NAT) by RELAT_1:90;
thus (IExec (I,P,s)) . f = ((Result ((P +* I),(Initialize (Initialized s)))) +* (s | NAT)) . f by SCMFSA8A:13
.= (Result ((P +* I),(Initialize (Initialized s)))) . f by A4, FUNCT_4:12
.= (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f by A2, EXTPRO_1:23 ; :: thesis: verum